Express In Simplest Form With A Rational Denominator.\[$\frac{5}{\sqrt{8}}\$\]Answer Attempt 1 Out Of 2\[$\square\$\]

Feb 26, 2025 by ADMIN 118 views

**Expressing a Rational Expression in Simplest Form with a Rational Denominator**

Introduction

In mathematics, a rational expression is a fraction that contains variables or constants in the numerator or denominator. When we are asked to express a rational expression in simplest form with a rational denominator, we need to simplify the expression by eliminating any radicals or fractions in the denominator. In this article, we will focus on expressing the rational expression $\frac{5}{\sqrt{8}}$ in simplest form with a rational denominator.

Understanding the Problem

The given rational expression is $\frac{5}{\sqrt{8}}$ . To express this expression in simplest form with a rational denominator, we need to eliminate the radical in the denominator. We can do this by multiplying the numerator and denominator by the radical in the denominator.

Step 1: Multiply the Numerator and Denominator by the Radical

To eliminate the radical in the denominator, we need to multiply the numerator and denominator by the radical in the denominator. In this case, we need to multiply the numerator and denominator by $\sqrt{8}$ .

$\frac{5}{\sqrt{8}} \cdot \frac{\sqrt{8}}{\sqrt{8}} = \frac{5\sqrt{8}}{8}$

Step 2: Simplify the Expression

Now that we have eliminated the radical in the denominator, we can simplify the expression by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF of $5\sqrt{8}$ and $8$ is $1$ , so we cannot simplify the expression further.

$\frac{5\sqrt{8}}{8}$

Conclusion

In this article, we have expressed the rational expression $\frac{5}{\sqrt{8}}$ in simplest form with a rational denominator. We did this by multiplying the numerator and denominator by the radical in the denominator and then simplifying the expression by dividing the numerator and denominator by their GCF. The final expression is $\frac{5\sqrt{8}}{8}$ .

Example 1: Expressing a Rational Expression with a Radical in the Denominator

Express the rational expression $\frac{3}{\sqrt{5}}$ in simplest form with a rational denominator.

Solution

To express the rational expression $\frac{3}{\sqrt{5}}$ in simplest form with a rational denominator, we need to multiply the numerator and denominator by the radical in the denominator.

$\frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$

Step 2: Simplify the Expression

Now that we have eliminated the radical in the denominator, we can simplify the expression by dividing the numerator and denominator by their GCF. In this case, the GCF of $3\sqrt{5}$ and $5$ is $1$ , so we cannot simplify the expression further.

$\frac{3\sqrt{5}}{5}$

Conclusion

In this example, we have expressed the rational expression $\frac{3}{\sqrt{5}}$ in simplest form with a rational denominator. We did this by multiplying the numerator and denominator by the radical in the denominator and then simplifying the expression by dividing the numerator and denominator by their GCF. The final expression is $\frac{3\sqrt{5}}{5}$ .

Example 2: Expressing a Rational Expression with a Radical in the Denominator

Express the rational expression $\frac{2}{\sqrt{3}}$ in simplest form with a rational denominator.

Solution

To express the rational expression $\frac{2}{\sqrt{3}}$ in simplest form with a rational denominator, we need to multiply the numerator and denominator by the radical in the denominator.

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

Step 2: Simplify the Expression

Now that we have eliminated the radical in the denominator, we can simplify the expression by dividing the numerator and denominator by their GCF. In this case, the GCF of $2\sqrt{3}$ and $3$ is $1$ , so we cannot simplify the expression further.

$\frac{2\sqrt{3}}{3}$

Conclusion

In this example, we have expressed the rational expression $\frac{2}{\sqrt{3}}$ in simplest form with a rational denominator. We did this by multiplying the numerator and denominator by the radical in the denominator and then simplifying the expression by dividing the numerator and denominator by their GCF. The final expression is $\frac{2\sqrt{3}}{3}$ .

Tips and Tricks

When expressing a rational expression in simplest form with a rational denominator, remember to:

Multiply the numerator and denominator by the radical in the denominator
Simplify the expression by dividing the numerator and denominator by their GCF
Check if the expression can be simplified further by dividing the numerator and denominator by their GCF

By following these tips and tricks, you can express rational expressions in simplest form with a rational denominator with ease.

Common Mistakes to Avoid

When expressing a rational expression in simplest form with a rational denominator, be careful not to:

Forget to multiply the numerator and denominator by the radical in the denominator
Simplify the expression incorrectly by dividing the numerator and denominator by the wrong GCF
Not check if the expression can be simplified further by dividing the numerator and denominator by their GCF

By avoiding these common mistakes, you can ensure that your rational expressions are expressed in simplest form with a rational denominator.

Conclusion

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables or constants in the numerator or denominator.

Q: What is the simplest form of a rational expression?

A: The simplest form of a rational expression is one in which the numerator and denominator have no common factors other than 1.

Q: How do I express a rational expression in simplest form with a rational denominator?

A: To express a rational expression in simplest form with a rational denominator, you need to multiply the numerator and denominator by the radical in the denominator and then simplify the expression by dividing the numerator and denominator by their greatest common factor (GCF).

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I find the GCF of two numbers?

A: To find the GCF of two numbers, you can list the factors of each number and find the largest factor that they have in common.

Q: What is the difference between a rational expression and a radical expression?

A: A rational expression is a fraction that contains variables or constants in the numerator or denominator, while a radical expression is an expression that contains a radical, such as a square root.

Q: Can I simplify a rational expression that has a radical in the denominator?

A: Yes, you can simplify a rational expression that has a radical in the denominator by multiplying the numerator and denominator by the radical in the denominator and then simplifying the expression by dividing the numerator and denominator by their GCF.

Q: What is the final form of a rational expression that has been simplified?

A: The final form of a rational expression that has been simplified is one in which the numerator and denominator have no common factors other than 1.

Q: Can I use a calculator to simplify a rational expression?

A: Yes, you can use a calculator to simplify a rational expression. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Q: What are some common mistakes to avoid when simplifying rational expressions?

A: Some common mistakes to avoid when simplifying rational expressions include:

Forgetting to multiply the numerator and denominator by the radical in the denominator
Simplifying the expression incorrectly by dividing the numerator and denominator by the wrong GCF
Not checking if the expression can be simplified further by dividing the numerator and denominator by their GCF

Q: How can I practice simplifying rational expressions?

A: You can practice simplifying rational expressions by working through examples and exercises in a textbook or online resource. You can also try simplifying rational expressions on your own by creating your own examples and exercises.

Q: What are some real-world applications of simplifying rational expressions?

A: Simplifying rational expressions has many real-world applications, including:

Calculating probabilities and statistics
Solving problems in physics and engineering
Working with financial data and investments
Creating and analyzing mathematical models

Conclusion

In this article, we have answered some common questions about expressing rational expressions in simplest form with a rational denominator. We have also provided tips and tricks for simplifying rational expressions and avoiding common mistakes. By following these tips and tricks, you can ensure that your rational expressions are expressed in simplest form with a rational denominator.